--- a/src/HOL/Library/Rounded_Division.thy Tue Oct 04 09:12:39 2022 +0000
+++ b/src/HOL/Library/Rounded_Division.thy Tue Oct 04 09:12:41 2022 +0000
@@ -8,14 +8,19 @@
begin
definition rounded_divide :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close> (infixl \<open>rdiv\<close> 70)
- where \<open>k rdiv l = (k + l div 2 + of_bool (l < 0)) div l\<close>
+ where \<open>k rdiv l = sgn l * ((k + \<bar>l\<bar> div 2) div \<bar>l\<bar>)\<close>
definition rounded_modulo :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close> (infixl \<open>rmod\<close> 70)
- where \<open>k rmod l = k - k rdiv l * l\<close>
+ where \<open>k rmod l = (k + \<bar>l\<bar> div 2) mod \<bar>l\<bar> - \<bar>l\<bar> div 2\<close>
lemma rdiv_mult_rmod_eq:
\<open>k rdiv l * l + k rmod l = k\<close>
- by (simp add: rounded_divide_def rounded_modulo_def)
+proof -
+ have *: \<open>l * (sgn l * j) = \<bar>l\<bar> * j\<close> for j
+ by (simp add: ac_simps abs_sgn)
+ show ?thesis
+ by (simp add: rounded_divide_def rounded_modulo_def algebra_simps *)
+qed
lemma mult_rdiv_rmod_eq:
\<open>l * (k rdiv l) + k rmod l = k\<close>
@@ -67,11 +72,32 @@
lemma zero_rmod_eq [simp]:
\<open>0 rmod k = 0\<close>
+ by (auto simp add: rounded_modulo_def not_less zmod_trivial_iff)
+
+lemma rdiv_minus_eq:
+ \<open>k rdiv - l = - (k rdiv l)\<close>
+ by (simp add: rounded_divide_def)
+
+lemma rmod_minus_eq [simp]:
+ \<open>k rmod - l = k rmod l\<close>
+ by (simp add: rounded_modulo_def)
+
+lemma rdiv_abs_eq:
+ \<open>k rdiv \<bar>l\<bar> = sgn l * (k rdiv l)\<close>
+ by (simp add: rounded_divide_def)
+
+lemma rmod_abs_eq [simp]:
+ \<open>k rmod \<bar>l\<bar> = k rmod l\<close>
by (simp add: rounded_modulo_def)
lemma nonzero_mult_rdiv_cancel_right:
\<open>k * l rdiv l = k\<close> if \<open>l \<noteq> 0\<close>
- using that by (auto simp add: rounded_divide_def ac_simps)
+proof -
+ have \<open>sgn l * k * \<bar>l\<bar> rdiv l = k\<close>
+ using that by (simp add: rounded_divide_def)
+ with that show ?thesis
+ by (simp add: ac_simps abs_sgn)
+qed
lemma rdiv_self_eq [simp]:
\<open>k rdiv k = 1\<close> if \<open>k \<noteq> 0\<close>
@@ -79,6 +105,18 @@
lemma rmod_self_eq [simp]:
\<open>k rmod k = 0\<close>
- by (cases \<open>k = 0\<close>) (simp_all add: rounded_modulo_def)
+proof -
+ have \<open>(sgn k * \<bar>k\<bar> + \<bar>k\<bar> div 2) mod \<bar>k\<bar> = \<bar>k\<bar> div 2\<close>
+ by (auto simp add: zmod_trivial_iff)
+ also have \<open>sgn k * \<bar>k\<bar> = k\<close>
+ by (simp add: abs_sgn)
+ finally show ?thesis
+ by (simp add: rounded_modulo_def algebra_simps)
+qed
+
+lemma signed_take_bit_eq_rmod:
+ \<open>signed_take_bit n k = k rmod (2 ^ Suc n)\<close>
+ by (simp only: rounded_modulo_def power_abs abs_numeral flip: take_bit_eq_mod)
+ (simp add: signed_take_bit_eq_take_bit_shift)
end